To find the 4th term of the geometric sequence $-8, -48, -288, \dots$, we need to determine the common ratio ($r$) and use the general formula for the $n$th term of a geometric sequence.

Step 1: Find the common ratio ($r$)

The common ratio $r$ is the ratio between consecutive terms:

$r = \frac{\text{second term}}{\text{first term}} = \frac{-48}{-8} = 6.$

Step 2: Use the general formula

The formula for the $n$th term of a geometric sequence is:

$a_n = a_1 \cdot r^{n-1},$

where:

• $a_n$ is the $n$th term,
• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the term number.

Step 3: Find the 4th term

Substitute the values into the formula:

• $a_1 = -8$,
• $r = 6$,
• $n = 4$.

$a_4 = -8 \cdot 6^{4-1} = -8 \cdot 6^3 = -8 \cdot 216 = -1728.$

Final Answer:

The 4th term is $-1728$.

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Would you like a more detailed explanation or have additional questions?

Related Questions:

1. What is the formula to find the sum of the first $n$ terms of a geometric sequence?
2. How do you determine if a sequence is geometric?
3. Can the common ratio be negative, and how does it affect the sequence?
4. What are some real-life examples of geometric sequences?
5. How do you find the 10th term of this sequence?

Tip:

Always double-check the common ratio by dividing consecutive terms to ensure accuracy!